What is the Bellman Equation actually telling?
For a Markov Decision Process $(\mathcal{S}, \mathcal{A}, P, R)$ (here $P(s, s') = \mathbb{P}(S_{t+1} = s' | S_t = s, A_t = a))$;, let us define the value of being in a certain state. That is,
$$v_\pi(s) = \mathbb{E}_{a_i \sim \pi, s_i \sim P}\left[\sum_{i=0}^\infty \gamma^{i+t}r(s_{t+i}, a_{i+t}) | S_t =s\right].$$
That is, the value of being in state $s$ at time $t$ is equal to the expected value of the discounted sum of future rewards, where the expectation is taken with respect to $\pi(\cdot | s)$ the action policy and the environment dynamics $P$.
We will now define
$$\sum_{i=0}^\infty \gamma^{i+t}r(s_{t+i}, a_{i+t}) = G_t$$
Now, we can rewrite this as
$$\mathbb{E}_{a_i \sim \pi, s_i \sim P}\left[G_t | S_t =s\right] =  \mathbb{E}_{a_i \sim \pi, s_i \sim P}\left[ r(s_t, a_t) + \gamma G_{t+1} | S_t =s\right].$$
The RHS is now in a nicer format and we can express it as
$$\mathbb{E}_{a_i \sim \pi, s_i \sim P}\left[G_t | S_t =s\right] = \mathbb{E}_{a_i \sim \pi, s_i \sim P}\left[ r(s_t, a_t) + \gamma v_\pi(s') | S_t =s\right],$$
where $s'$ is the state that we transition into at time $t+1$. Note that $s'$ is a random variable and we are taking the expectation over this according to the action policy and the environment dynamics -- this is because the state we transition into depends first on the action we select and then the environment dynamics $P$.
You can see that we now have
$$v_\pi(s) = \mathbb{E}_{a_i \sim \pi, s_i \sim P}\left[ r(s_t, a_t) + \gamma v_\pi(s') | S_t =s\right].$$
This is the Bellman equation (at least a form of it) and it expresses a recursive relationship between the values of states. That is, the value of being in state $s$ is equal to the expected immediate reward from being in this state plus the value of being in the state that we transition into. This relationship is useful in Reinforcement Learning as many algorithms use this equation to form update rules to approximate the value/state-action value function of the MDP, such as the  SARSA algorithm, so it is more than just a philosophy, it is the driving force behind many of the RL algorithms.
Now, when I say at least a form of it, that is because in RL it is also common to see a Bellman equation for the state action value function
$$q_\pi(s, a) = \mathbb{E}_{a_i \sim \pi, s_i \sim P}\left[ G_t | S_t = s, A_t = a\right] = \mathbb{E}_{a_i \sim \pi, s_i \sim P}\left[ r(s_t, a_t) + v_\pi(s') | S_t = s, A_t = a\right];$$
noting that the value function $v$ is the expectation of $q$ over the action space, i.e. we marginalise out the action we condition on -- thus there is the recursive relationship.
As pointed out in the comments for this question, it is also worth noting that Bellman equations originated in Dynamic Programming, which exist to solve planning problems such as the knapsack problem.